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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $Z \\subseteq X$ be a [[closed set|0ZCK8DQYN1]].\nA point $\\eta \\in Z$ is a *generic point* of $Z$ if the [[closure|0QEAJB91GZ]] of $\\set{\\eta}$ is $Z$.\nA generic point of $X$ is a generic point of the closed set $X$.\n\n## Remarks\n\nA closed set with a generic point is irreducible: it is not the union of two smaller closed sets.\nIn a [[scheme|5A35TYZD02]], every irreducible closed subset has a unique generic point, so the underlying space of a scheme is a [[sober space|4N4DNDR22Z]].\nExample: in $X = \\operatorname{Spec} \\Z$, the closure of $\\set{(0)}$ is $V(0) = X$, because $(0)$ is contained in every prime ideal.\nSo $(0)$ is the generic point of $X$.\nNon-example: the point $(p)$, for a prime $p$, is not a generic point of $X$, since $\\set{(p)}$ is already closed.\nNon-example: $X$ itself has no generic point when $X = \\operatorname{Spec} k[x, y]/(xy)$ for a field $k$, since $X$ is the union of the two lines $V(x)$ and $V(y)$.\n",
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